Analysis of fractal radiation patterns from concentric-ring hexagonal arrays
These hexagonal arrays are becoming increasingly popular, especially for their applications in the area of wireless communications. The overall objective of this article has been to use the theoretical foundation developed for the analysis of radiation patterns and design of the hexagonal arrays. A...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2005
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| Цитувати: | Analysis of fractal radiation patterns from concentric-ring hexagonal arrays / A. Boualleg, N. Merabtine, M. Benslama // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 4. — С. 91-94. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1215712017-06-15T03:03:45Z Analysis of fractal radiation patterns from concentric-ring hexagonal arrays Boualleg, A. Merabtine, N. Benslama, M. These hexagonal arrays are becoming increasingly popular, especially for their applications in the area of wireless communications. The overall objective of this article has been to use the theoretical foundation developed for the analysis of radiation patterns and design of the hexagonal arrays. A technique has been developed for the analysis of radiation patterns from concentric ring arrays. A family of functions, known as generalized Weierstrass functions, has been shown to play a key role in the theory of fractal radiation pattern analysis. 2005 Article Analysis of fractal radiation patterns from concentric-ring hexagonal arrays / A. Boualleg, N. Merabtine, M. Benslama // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 4. — С. 91-94. — Бібліогр.: 4 назв. — англ. 1560-8034 PACS 84.40.Ba http://dspace.nbuv.gov.ua/handle/123456789/121571 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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These hexagonal arrays are becoming increasingly popular, especially for their applications in the area of wireless communications. The overall objective of this article has been to use the theoretical foundation developed for the analysis of radiation patterns and design of the hexagonal arrays. A technique has been developed for the analysis of radiation patterns from concentric ring arrays. A family of functions, known as generalized Weierstrass functions, has been shown to play a key role in the theory of fractal radiation pattern analysis. |
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Article |
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Boualleg, A. Merabtine, N. Benslama, M. |
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Boualleg, A. Merabtine, N. Benslama, M. Analysis of fractal radiation patterns from concentric-ring hexagonal arrays Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Boualleg, A. Merabtine, N. Benslama, M. |
| author_sort |
Boualleg, A. |
| title |
Analysis of fractal radiation patterns from concentric-ring hexagonal arrays |
| title_short |
Analysis of fractal radiation patterns from concentric-ring hexagonal arrays |
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Analysis of fractal radiation patterns from concentric-ring hexagonal arrays |
| title_fullStr |
Analysis of fractal radiation patterns from concentric-ring hexagonal arrays |
| title_full_unstemmed |
Analysis of fractal radiation patterns from concentric-ring hexagonal arrays |
| title_sort |
analysis of fractal radiation patterns from concentric-ring hexagonal arrays |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/121571 |
| citation_txt |
Analysis of fractal radiation patterns from concentric-ring hexagonal arrays / A. Boualleg, N. Merabtine, M. Benslama // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 4. — С. 91-94. — Бібліогр.: 4 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT bouallega analysisoffractalradiationpatternsfromconcentricringhexagonalarrays AT merabtinen analysisoffractalradiationpatternsfromconcentricringhexagonalarrays AT benslamam analysisoffractalradiationpatternsfromconcentricringhexagonalarrays |
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2025-07-08T20:08:34Z |
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2025-07-08T20:08:34Z |
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1837110726517325824 |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 91-94.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
91
PACS 84.40.Ba
Analysis of fractal radiation patterns
from concentric-ring hexagonal arrays
A. Boualleg, N. Merabtine, M. Benslama
Laboratory of Electromagnetism and Telecommunications LET, Department of Electronics,
University Mentouri Constantine, Algeria
E-mail:bouadzdz@yahoo.fr; na_merabtine@hotmail.com; malekbenslama@hotmail.com
Abstract. These hexagonal arrays are becoming increasingly popular, especially for their
applications in the area of wireless communications. The overall objective of this article
has been to use the theoretical foundation developed for the analysis of radiation patterns
and design of the hexagonal arrays. A technique has been developed for the analysis of
radiation patterns from concentric ring arrays. A family of functions, known as
generalized Weierstrass functions, has been shown to play a key role in the theory of
fractal radiation pattern analysis.
Keywords: fractal antenna arrays, fractal antenna radiation patterns, low side-lobe
antennas radiation patterns.
Manuscript received 30.09.05; accepted for publication 25.10.05.
1. Introduction
The name “fractal”, from the Latin “fractus” meaning
broken, was given to highly irregular sets by Benoit
Mandelbrot in his foundational essay in 1975 [1]. Since
then, fractal geometry has attracted widespread, and
sometimes controversial, attention. The subject has
grown on two fronts: on the one hand, many “real
fractals” of science and nature have been identified. On
the other hand, mathematics that is available for
studying fractal sets, a lot of which has its roots in
geometric measure theory, has developed enormously
with new tools emerging for fractal analysis. This paper
concerned with mathematics of fractals and application
to the antenna theory [2].
2. Theory
The standard hexagonal arrays are formed by placing
elements in equilateral triangular grid with spacings d.
These arrays can also be rounded by several concentric
six-element circular arrays of different radii [3, 4]. The
resulting expression for the hexagonal array factor, in a
normalized form, is given by [2].
( )
( )[ ]
∑∑∑
∑∑∑
= = =
= = =
+−
+
+
= P
p
P
m n
pmn
P
p
P
m n
krj
pmn
P
II
eII
F
pmnpmnpm
1 1
5
0
0
1 1
5
0
cossin
0
,
αϕϕθ
ϕθ ,
(1)
where
( ) ( )11 22 −−−+= mpmpdrpm ,
( )
32
1
cos
2222
1
2
np
dpr
mdpdr
pm
pm
pmn +
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡ −−+
= −ϕ ,
( )pmnpmpmn kr ϕϕθα −−= 00 cossin
and P is the number of concentric hexagons in the array.
Hence, the total number of elements contained in an
array with P hexagons is
( ) 113 ++= PPN P .
At this point: we investigate the possibility that
useful designs for hexagonal arrays may be realized via
a construction process based on the recursive application
of a generating subarray. To demonstrate this, suppose
we consider the uniformly excited six-element circular
generating subarray of the radius 2/λ=r , shown in
Fig. 1.
This particular value of the radius was chosen so that
these six elements in the array correspond to the vertices
of a hexagon with half-wavelength sides (i. e., 2/λ=r ).
Consequently, the array factor associated with this six-
element generating subarray may be shown to have the
following representation:
( ) ( )[ ]∏∑
= =
+−−
=
P
p n
j
PP nn
P
eF
1
6
1
cossin1
6
1, αϕϕθπδϕθ , (2)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 91-94.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
92
Fig. 1. The geometry for a uniformly excited six-element
circular subarray generator of the radius 2/λ=r .
Fig. 2. The first four stages in the construction of a hexagonal
array via the generating subarray illustrated in Fig. 1, with the
expansion factor 2=δ .The element locations correspond to the
vertices of the hexagons.
where
( )
3
1 πϕ −= nn , ( )nn ϕϕθπα −−= 00 cossin .
The array factor expression given in (2) may also be
written in the form
( ) ( )∏∑
= =
−
=
P
p n
j
PP n
P
eF
1
6
1
,1
6
1, ϕθψδϕθ , (3)
where
( ) ( ) ( )[ ]nnn ϕϕθϕϕθπϕθψ −−−= 00 cossincossin, .
We will first examine the special case where the
expansion factor of the recursive hexagonal array is
assumed to be unity, i.e., 1=δ . Under these
circumstances, equation (3) reduces to
( ) ( )
P
n
j
P neF
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
= ∑
=
6
1
,
6
1, ϕθψϕθ . (4)
These arrays increase in size at a rate that obeys the
relationship
( ) ( )plP PPN δ−++= 113 ,
where plδ represents the Kronecker delta function
defined by [2]
⎩
⎨
⎧
≠
=
=
10
1,1
P
P
plδ .
In other words, every time this fractal array evolves
from one stage to the next, the number of concentric
hexagonal subarray contained in it increases by one.
The second special case of interest to be considered
in this section results when a choice of 2=δ is made.
Substituting this value of δ into (3) yields an expression
for the recursive hexagonal array factor given by
( ) ( )∏∑
= =
−
=
P
p n
j
PP n
P
eF
1
6
1
,2 1
6
1, ϕθψϕθ , (5)
where
( ) ( )[ ]1221223 11 −−−= −− PPPP
PN .
Clearly, by comparing PN (for 2=δ ) with PN
(for 1=δ ), we conclude that these recursive arrays will
grow at a much faster rate than those generated by a
choice of 1=δ . The representations of the first four
stages in the construction process of these arrays are
illustrated in Fig. 2, where the element locations
correspond to the vertices of the hexagons.
Fig. 2 indicates that the hexagonal arrays resulting
from the recursive construction process with 2=δ have
some elements missing, i. e., they are thinned.
It is interesting to look what happens with these
arrays when an element with two units of current is
added to the center of the hexagonal generating subarray
shown in Fig. 1. Under these circumstances, the
expression for the array factor given in (3) must be
modified in the following way:
( ) ( )∏ ∑
= = ⎪⎭
⎪
⎬
⎫
⎪⎩
⎪
⎨
⎧
+=
−
P
p n
j
PP n
p
eF
1
6
1
,1
2
8
1, ϕθψδϕθ . (6)
Plots of a several radiation patterns calculated from
(6) with 1=δ and 2=δ are shown in Figs 3 and 4,
respectively. It is evident from Fig. 4 that the radiation
patterns for these arrays have no side lobes.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 91-94.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
93
Fig. 3. Plots of the far field radiation patterns produced by a series of four (P=1, 2, 3, and 4) fully populated hexagonal arrays
generated with the expansion factor 1=δ .
Fig. 4. Plots of the far field radiation patterns produced by a series of four (P =1, 2, 3, and 4) fully populated hexagonal arrays
generated with the expansion factor 2=δ .
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 4. P. 91-94.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
94
These plots indicate that a further reduction in side-lobe
levels may be achieved by including a central element in
the generating subarray of Fig. 1.
3. Conclusion
The research in the area of fractal antenna has recently
yielded a rich class of new designs for antenna elements
as well as arrays.
The essential property of hexagonal arrays have
some elements missing, i. e., they are thinning. This is a
potential advantage of these arrays from the design point
of view, since they may be realized with fewer elements.
Another property of these arrays is that they possess low
side-lobe levels at 2/πϕ = . Finally, it should be noted
that the compact product form of the array factor can be
obtained for some particular cases. This offers a
significant advantage in terms of computational
efficiency, especially for large arrays, and may be
exploited to develop rapid beam-forming algorithms.
References
1. B.B. Mandelbrot, The fractal geometry of nature.
San Francisco (1975).
2. D.H. Werner, R.L. Haupt and P.L. Werner, Fractal
antenna engineering: The theory and design of fractal
Antenna Arrays // IEEE Antennas and Propagation
Magazine 41, No 5, p. 37-59 (1999).
3. V.F. Kravchenko, and V.M. Masyuk, Analysis and
synthesis of multiband antenna arrays //
Electromagnetic waves and electronic systems 9, Nos
3-4 (2004).
4. V.F. Kravchenko, and V.M. Masyuk, The ring fractal
antenna arrays // Electromagnetic waves and
electronic systems 9, No 5 (2004).
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